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Question 30098: Intersection and Conditioning-Event Probabilities

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0059-30098 · q-20261008-advance-20261010-r04-conditional-probability-0023

Original question and givens

S={0,…,3}, F=P(S), masses=[22, 1, 2, 0], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.

  • G001 — S={0,…,3}, F=P(S), masses=[22, 1, 2, 0], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-density10-conditional-masses-pair, density10_conditional_masses_pair, finite_normalized_probability]

After

derived = [[P_intersection, P_B]]

Conditions: none

Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources.

Step identifiers and dependencies

Rule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001

S002 · calculation

Before

Conditional ORIGINAL ALL numerator weights sum := sum(22, 1, 2, 0)

After

Conditional ORIGINAL ALL numerator weights sum = 25

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

source roles = [[22, 1, 2, 0], 25]

After

derived = [4, [[0, 22/25], [1, 1/25], [2, 2/25], [3, 0/1]], 25]

Conditions: none

On S=Fin(n) with its full powerset,each singleton has mass m_i/D,all m_i>=0 and total D>0.These FOUR families REQUIRE an actual normalized law;reject an invalid law rather than return No and repair its weights.

Step identifiers and dependencies

Rule: Conditional_FULL_normalized_finite_law_not_candidate_repair · Uses: G001, S002

S004 · implies

Before

source roles = [B, [0, 1], 4]

After

derived = [[0, 1]]

Conditions: none

FIRST require raw length<=16 and each SAFE index in Fin(n),THEN deduplicate and sort numerically.Repeated listings do not multiply mass;17 raw copies still exceed the source bound.

Step identifiers and dependencies

Rule: Conditional_RAW_event_bound_BEFORE_dedup_and_Fin_index_typing · Uses: G001, S003

S005 · calculation

Before

Conditional B ORIGINAL included index0 running0 := sum(0, 22)

After

Conditional B ORIGINAL included index0 running0 = 22

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Conditional B ORIGINAL included index1 running1 := sum(22, 1)

After

Conditional B ORIGINAL included index1 running1 = 23

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [Conditional B, [0, 1], [22, 1, 2, 0]]

After

derived = [23]

Conditions: none

For each normalized event index take its ORIGINAL slot exactly once.An empty event has sum0;do not invent uniform weights or change alignment.

Step identifiers and dependencies

Rule: Conditional_COMPLETE_subset_mass_in_ORIGINAL_slots · Uses: G001, S006

S008 · calculation

Before

Conditional ORIGINAL B mass over D := div(23, 25)

After

Conditional ORIGINAL B mass over D = 23/25

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [[0, 1], [22, 1, 2, 0], 25]

After

derived = [23, 23/25, Yes]

Conditions: none

P(B)>0 means b>0,not merely B nonempty.A nonempty event containing only null labels cannot be conditioned on.Divide by b ONLY under the given positive-mass premise.

Step identifiers and dependencies

Rule: Conditional_STRICT_positive_B_mass_not_merely_nonempty · Uses: G001, S008

S010 · implies

Before

source roles = [A, [0, 2], 4]

After

derived = [[0, 2]]

Conditions: none

FIRST require raw length<=16 and each SAFE index in Fin(n),THEN deduplicate and sort numerically.Repeated listings do not multiply mass;17 raw copies still exceed the source bound.

Step identifiers and dependencies

Rule: Conditional_RAW_event_bound_BEFORE_dedup_and_Fin_index_typing · Uses: G001, S009

S011 · calculation

Before

Conditional intersection ORIGINAL included index0 running0 := sum(0, 22)

After

Conditional intersection ORIGINAL included index0 running0 = 22

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [Conditional intersection, [0], [22, 1, 2, 0]]

After

derived = [22]

Conditions: none

For each normalized event index take its ORIGINAL slot exactly once.An empty event has sum0;do not invent uniform weights or change alignment.

Step identifiers and dependencies

Rule: Conditional_COMPLETE_subset_mass_in_ORIGINAL_slots · Uses: G001, S011

S013 · calculation

Before

Conditional ORIGINAL intersection mass over D := div(22, 25)

After

Conditional ORIGINAL intersection mass over D = 22/25

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [[0, 2], [0, 1], [0], [22, 1, 2, 0], 25]

After

derived = [22/25, 23/25]

Conditions: none

These TWO fields are P(A intersection B) and P(B),each divided by D,not the conditional ratio i/b.Keep their declared order.

Step identifiers and dependencies

Rule: Conditional_MASSES_pair_separate_unconditional_denominator_D · Uses: G001, S013

Reverse checks

  • S014 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S013 · premise_and_result_audit — Recompute Conditional ORIGINAL intersection mass over D with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S011 · premise_and_result_audit — Recompute Conditional intersection ORIGINAL included index0 running0 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S009 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S008 · premise_and_result_audit — Recompute Conditional ORIGINAL B mass over D with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S006 · premise_and_result_audit — Recompute Conditional B ORIGINAL included index1 running1 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Conditional B ORIGINAL included index0 running0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S003 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S002 · premise_and_result_audit — Recompute Conditional ORIGINAL ALL numerator weights sum with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.

Result

(22/25, 23/25)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 23ee8e9586ee81f0132028e985091046f08bcb53e0d4fd3b5a272e10d9f188cc
  • candidate record SHA-256: 7a6e72f5faa7f14bc869db0c788bc1c86d27051a78bcd96025bbf7c73c9093b5
  • semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
  • question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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